En ma fin gît mon commencement.
— Mary, Queen of Scots
The ever-kindling fire of eternal inflation grants the universe an appearance of self-sufficiency. Our observable universe emerges as a thermalized bubble from a still-inflating bulk that forever outruns it, spawning innumerable other bubble universes in endless array. But if inflation runs on interminably ahead of us, might it not also have gone on similarly before us, without a first moment and owing no physical account of its origin? Alexander Vilenkin framed this possibility as an irresistible force confronting an immovable obstruction.1 The immovable obstruction is a four-page theorem published in 2003 by Arvind Borde, Alan Guth, and Vilenkin himself (BGV).2 I propose to examine the bonds by which this theorem binds our universe: what it claims, what it demonstrates, and where, weighed in the balance, it is found wanting. This last question will occupy most of our time, for the strength of a theorem is measured not just by the prison doors it clangs shut, but by the hidden passages it leaves open.
What is claimed
The BGV theorem rests on a solitary inequality that speaks doom over mild-mannered geodesics. Trace into the past any timelike or null geodesic representing the worldline of a possible observer or light ray. Along such a path, an average rate of expansion, Hav, may be defined that is the mean of the Hubble parameter. It is measured over proper time for timelike geodesics and over the affine parameter for null geodesics.3 When this average is positive, which is to say, when
Hav > 0,
the theorem shows that the past length of the relevant geodesic is finite and the geodesic cannot be extended backward indefinitely. If the clock is run backward, the geodesic will come to a stop in finite proper time for massive particles or after finite affine length for light rays. Were this to happen along even one such geodesic, the spacetime containing it would be, by definition, past-incomplete.
It is the poverty of the theorem’s hypotheses that gives it such richness of reach. There are no field equations. General relativity may hold or it may fail. Mass-energy can behave as strangely as you please without making any difference, for there are no strong, weak, or null energy conditions to satisfy.4 By contrast, the conclusions of the classical singularity theorems of Hawking and Penrose flow from the strong or null energy conditions.5 With the advent of inflationary cosmology, the strong condition was violated intentionally, and the null condition, which classical slow roll inflation left intact, was breached only in eternal inflation by quantum vacuum fluctuations that allow local increases in the expansion rate. The Borde–Vilenkin theorems that were predecessors of BGV invoked the weak energy condition, which also was violated by chaotic eternal inflation driving the expansion rate upward.6 The BGV theorem dispenses with all of this, relying solely on kinematics and eschewing dynamics. Its result is purely geometric, focusing on the way that congruences of trajectories spread without regard for the cause of their divergence. It makes no demand for homogeneity or isotropy, nor even that the expanding congruence fill all of spacetime. The only indispensable condition is that a smooth flow of worldlines, one passing through each event, be definable along the geodesic singled out for examination.
While its conditions are simple, the BGV theorem has limitations and there are some common confusions about it that need to be addressed. We start by drawing a distinction between classical singularities and the past-incompleteness that is the conclusion when the theorem’s conditions are satisfied; they are not the same thing. What BGV shows is that inflating regions have boundaries beyond which no inflationary description can be continued, and this is all that past-incompleteness means. The theorem is silent about what lies on the other side. It could be a quantum nucleation event, a classical singularity, or even a regime governed by unknown physics. The conservative conclusion is that inflation does not supply a complete past, so new physics of some sort must provide the boundary conditions. Furthermore, as we have seen, the BGV theorem is about individual geodesics, and different geodesics can, of course, end at different times. This means that while inflationary spacetimes are past-incomplete, there need not be a tidy past hypersurface assembled from the endpoints. Such a spacetime has a genuine edge to it, but the BGV theorem does not require it to be smooth.
How it is demonstrated
The BGV proof involves a kind of mathematical misdirection: a seemingly dynamical quantity, namely a measured expansion rate along a worldline that appears to depend on the entire history of the motion, turns out instead to be a total derivative, so its integral along the worldline is fixed by its endpoints and indifferent to its history.7
Consider first the easiest case, a homogeneous, isotropic, spatially flat universe with a scale factor, a(t). Let λ represent the affine parameter that serves as a photon’s mileage marker. Given this metric, any null geodesic satisfies dλ∝a(t) dt, thus tying each increment of affine parameter to an increment of cosmic time through the scale factor. This means that if a(t) tapers off fast enough as time goes backward, the integral ∫a(t) dt will converge and the affine length will be finite. As an illustration, consider null signals traveling along such a geodesic. They will be increasingly blueshifted as they are traced into the past, with their frequency scaling as 1a(t). As the signals get bluer, the affine parameter accrues ever more slowly, so if the expansion is sufficiently rapid, even an infinite stretch of cosmic time purchases only a finite affine length.
To free this result from the special assumptions characteristic of flat universes, the BGV theorem defines the expansion rate in terms of relative motion. To see how this works, let O be the geodesic worldline of any observer, traced backward in time, and let a smooth flow (congruence) of comoving test particles travel through the spacetime around it. When this geodesic passes successive test particles, we can measure their recession, and from this we can infer a Hubble rate that is the radial separation velocity divided by the separation distance, i.e., H = ΔurΔr. This quantity expresses how fast neighboring trajectories are drifting apart. If we now let uμ be the spacetime four-velocity of the comoving test particle that O is passing (pointing along the direction of cosmic time in the local rest frame) and vμ be the spacetime four-velocity of the observer, tilted away from uμ by however fast O is cutting through the flow, then the angle of tilt, so to speak, is measured by the factor γ = uμvμ between the test particle’s four-velocity and the observer’s. For a timelike observer, γ is just the familiar Lorentz factor for the observer’s speed through the comoving frame. In the case of a photon, however, γ = dtdτ measures how much clock time elapses per unit of its affine parameter. Time enters the null case through the cosmic flow because a photon does not have a clock, but every marker it passes does. Proper time thus provides the geodesic’s measure for a timelike observer, and the affine parameter does the same job for a light ray.
We now see how the misdirection is resolved. Defined in terms of relative motion, H collapses into a total derivative:
H = ddτ F(γ(τ)), where F(γ) = γ−1,null observer,12lnγ + 1γ − 1,timelike observer.
This gives us everything we need. Since F is positive, if we integrate along the geodesic from a point in the past to a reference point we get
τf∫τiH dτ = F(γf) − F(γi) ≤ F(γf).
Note that the velocity at a single reference point, τf, suffices to constrain the integral so that, regardless of how irregular or inhomogeneous the expansion is, or how indifferent the expansion is to Einstein’s equations, the integrated expansion must be finite along the geodesic. This balance cannot be exceeded.
What remains is just arithmetic. Suppose the average of H over a geodesic is a positive number, Hav. This means that the average, when multiplied by the geodesic’s length, equals the value of the bounded integral given above, in which case
τf − τi ≤ F(γf)Hav<∞.
An infinite past is thus incompatible with a positive average expansion. In the null case, which is simplest, normalization is straightforward because F(γf) = γf−1 by definition, and this is just dτdt at the reference point. If we calibrate the affine parameter by making one unit of affine length equal to one unit of cosmic time at the reference point, this sets F(γf) = 1, and 1Hav is then a bound on past affine length. Moving to the timelike case, the observer’s speed climbs toward the light barrier if we trace it backward relative to the comoving particles, and the slowness F(γ) sinks toward zero. This barrier is never reached because no point of a timelike worldline moves at the speed of light. Nor is the reserve of slowness ever exhausted, for its floor is never attained. Nonetheless, the reserve is finite and is drawn down at a positive average rate, so it can last only a finite stretch of proper time. This is where the inequality comes from. The worldline ends in finite proper time with the light barrier never attained. The past itself is running out, not the slowness. That is why we cannot trace the expanding flow of worldlines backward forever. The dwindling reserve can sustain only a finite interval, and when it is gone, there is no “when” for the geodesic to occupy.
Precision requires us to state two qualifications. The first is that, for a strictly comoving observer at rest in the flow, there is no relative motion and γ never changes. Because γ is constant, there is nothing that varies for H = ddτ F(γ(τ)) to track. The value of H is what an observer would determine from the markers he passes, but an observer at rest in the flow does not pass any. Even if the derivative could be taken, no bound would appear because F(γ) grows without limit as the relative velocity disappears, and infinity sets no bounds. There is no way to bring the hypothesis of the theorem to bear. This is not to say that comoving observers do not see universal expansion; they do. But the H in question is not the recession of the heavens. What it measures is the expansion along the observer’s own motion through the flow, and since he is carried by that flow, there is no relative motion. There is thus no finite bound on the past in this case, nor should there be: infinite proper time into the past is possible for comoving worldlines in an inflating spacetime. The theorem addresses null geodesics and noncomoving timelike worldlines, so idealized riders of the flow are exempt from it. The second qualification is that the smooth flow need exist only along the geodesic, O. Anywhere else, the worldlines of this congruence may cross or focus without affecting the argument. With these two qualifications in mind, the theorem is otherwise perfectly general and applies to cyclic models as well. Given a cyclic model in which each cycle expands more than it contracts, there is an average growth in volume, so Hav > 0 over any collection of cycles. In such instances, the theorem is as applicable to an infinite cyclic history as it is to smooth inflation itself.
Where it is found wanting
Every theorem, no matter how far-reaching and general, is limited in scope by the hypothesis that must be satisfied for it to apply: if such and such conditions are met, then the following consequences ensue. The BGV theorem is touted as showing that the universe cannot be past-eternal, but this consequence rides on the back of a hypothesis, and the universe might be past-eternal if the hypothesis is not satisfied.
The BGV theorem is predicated on a sign condition, Hav > 0, and a sign condition has three cases. An averaged expansion rate may be positive, negative, or zero, and the BGV theorem addresses only the first. Whether the cosmos is past-eternal must also be addressed along paths following mathematical road signs that the theorem ignores. The claim that no evolving universe can be past-eternal requires all three cases to receive the same verdict that the theorem pronounces on only one.
Take the second road, Hav < 0, then: a universe with a past that is, on average, contracting is the native territory of bouncing and cyclic cosmologies that try to thread an infinite past through endless crunches and rebounds. The BGV theorem is silent when the average Hubble parameter is negative. Even so, this case is constrained by a number of results, some quite old and others quite new. When strong or null energy conditions are invoked, the classical singularity theorems of Penrose and Hawking apply. For bouncing models, Tolman showed in the 1930s that an accumulation of entropy prohibits endless identical cycling, forcing the cycles successively to change in size and duration. This is the classical thermodynamic obstacle to a past-eternal cycling of the universe.8 To prevent the entropy density from increasing without bound over endless cycles, the volume of the universe must itself grow without bound. But this restores net expansion, in which case the BGV theorem becomes applicable again. This entropic barrier to endless cycles is further extended by Wall’s quantum singularity theorem via the generalized second law, which holds even when classical energy conditions fail.9 Finally, Kinney and Stein have shown that any bouncing cosmology that dissipates accumulating entropy by net growth in the scale factor has to be geodesically past-incomplete, their chief example being the cyclic slow-contraction model of Anna Ijjas and Paul Steinhardt.10 The bottom line is that there is no known contracting or bouncing cosmology that circumvents all these results. Some advocates of loop and matter-bounce cosmologies still struggle against the conclusion that past-eternality has been ruled out. In doing so, they try to exploit violations of the classical energy conditions and the semiclassical scaffolding of Wall’s argument. But such end-runs do not help when it comes to Tolman’s thermodynamic argument or the results of Kinney and Stein. The quantum evolution of the universe preserves information because the governing operator is unitary. Moreover, unitary evolution through the bounce is championed by bounce theorists. There is thus no physical mechanism in a bounce that eliminates the cycle-to-cycle accumulation of correlations and coarse-grained entropy. Simply asserting that entropy resets with each bounce is insufficient without giving a clear account of how the entropy disappears and the associated coarse-graining changes. Any objection that a coarse-grained reset at every bounce could preserve past-eternality is therefore just a plea for mechanism-free miracles, world without end. The second, contracting, road is thus blocked, not by a single result taken in isolation, but by the accumulation of many. A single boulder is an inconvenience, but an avalanche closes the way.
The third and last road we must consider is that of Hav = 0. It requires an infinite asymptotic stillness that gives rise to a universe. The idea is that the universe was at rest through an eternal past, existing as a latent cosmic seed that only sprouted a finite time ago, or a cosmic egg that, in comparison with its infinite history, only recently hatched. When the average expansion rate of the universe is considered over this infinite asymptotic past, it is zero, and so the BGV theorem does not apply. This strategy, invoking a quasi-static cosmological seed, undergirds the emergent-universe model proposed by Ellis and Maartens.11 Through consideration of a related static-seed model, Audrey Mithani and Alexander Vilenkin demonstrated that this geometric escape from the BGV theorem has a quantum-mechanical vulnerability.12 By solving the Wheeler–DeWitt equation in minisuperspace13 for a closed, classically stable model of the simple harmonic universe, they showed its oscillating seed was unstable. Its probability of tunneling to vanishing size is nonzero per unit of clock time. An observer of the infinite past would already have seen this collapse of the cosmic seed with probability one. Such cosmic seeds do not keep forever; they must either sprout within a finite time or go bad.
Rather than sprouting from a nascent quantum seed, Lesnefsky, Easson, and Davies have constructed “loitering” universes in classical spacetime that have scale factors that approach a nonzero constant asymptotically in the past. This renders them geodesically complete despite having a Hubble parameter that always remains positive. Such models have been promoted as counterexamples to the BGV theorem that offer a fourth road.14 Closer examination, however, shows this portrayal to be mistaken. The bounded integral of H is distributed along complete geodesics over an infinite past, so its average falls to zero. The spacetime thereby constructed is a variation on the third road masquerading as a fourth. Kinney and various collaborators have shown this in classical general relativity by refining the averaged condition.15
More can be said by shifting our focus from the kinematics of geodesics to the decay theory of quantum states. Mithani and Vilenkin’s calculation was done for one potential using one approximation, and their conclusion was correspondingly narrow in application. The lesson, however, can be generalized if one focuses on quantum dynamics rather than on any particular cosmological model.16 I summarize this argument in an appendix, but let me expand on its significance.
If we set aside the tunneling picture of the Mithani–Vilenkin proof, three distinct things remain. A cosmic seed is a quantum system with a stable ground state, so its Hamiltonian is bounded below. But it cannot be stationary, because a seed that never changes through time would never sprout or do anything at all. Finally, since the seed decays into the continuum of expanding geometries it is supposed to produce, its spectral weight must be spread over that continuum with an absolutely continuous density. Given these three conditions, the survival probability, S(t), of the seed state being found in its initial configuration at a later time t is
S(t) = |⟨ψ,e−iHtψ⟩|2 = ∞∫0e−iEt dμψ(E)2.
In the 1940s, the Russian physicists Vladimir Fock and Nikolay Krylov showed that the squared Fourier transform of an absolutely continuous spectral measure,17 by the Riemann–Lebesgue lemma, vanishes at large times, driving this survival probability to zero.18 A further theorem by another Russian physicist, Leonid Khalfin, shows that the decay is slower than exponential (it follows the generic form of a power law), but it makes no difference to the outcome.19 The descent is gentler than one might expect, but its floor is zero. More interesting for our purposes is the fact that the survival probability of the state is an even function of time. Past and future decay are symmetric, regardless of how the state was prepared. Stated plainly, this means that quasi-static (metastable) states are no more able to have survived an infinite past in their initial seed form than they could persist in that form forever into the future.
Arguments like this have a long and honorable history, going back at least as far as the pre-Socratic philosophers. Parmenides would have reality to be changeless, eternal, and static; Heraclitus would have it as fire, forever changing and never twice the same. The reality of Parmenides is eternally constant and brooks no variance; Heraclitean reality, by contrast, is perpetually novel and fertile. By analogy, a quantum seed in an exact eigenstate of its Hamiltonian is eternal and forever still in the sense that it evolves only by a phase and thus yields no transitions, no direction for time, and certainly no expanding universe. Such a seed does not lead to us and cannot be our past. A quantum seed capable of giving irreversible rise to an expanding cosmos, on the other hand, is coupled to the continuum, and its Hamiltonian is bounded below so that, while it can and will be productive, it cannot be past-eternal. The wall of separation between bound or recurrent seeds that have persisted and will persist forever, and continuum-coupled expanding seeds that are not eternal, is absolute. There is no tertium quid of eternally persistent and irreversibly productive seeds. What we see, then, is that the price of productivity is instability, and the price of instability is a finite past. Mithani and Vilenkin demonstrated this fact for one potential, but the spectral no-go theorem secures its general basis.
The foundational premises
The spectral no-go theorem I summarized is purchased at the price of two essential assumptions, one that requires defending against a likely objection, and a second that derives from the very job the cosmic seed is being asked to do. First, the theorem assumes that cosmic seed models have an internal time (a clock) telling when one state succeeds another, because the survival probability, S(τ), requires it. The central problem here is that, outside the homogeneous minisuperspace in which all such seed models are constructed, quantum gravity is widely taken to be timeless. Applied to the universe as a whole, the Wheeler–DeWitt equation describes a total state that does not evolve, so when everything is considered together, nothing happens. This is the so-called problem of time in quantum gravity. It is thus possible that a determined defender of past-eternality will try to use this problem as a basis for denying that the cosmic seed evolves at all. That would be a mistake.
Denying that the cosmic seed evolves does not give the past-eternalist what he wants. There are three reasons for this. First, past-eternal seed models are formulated in minisuperspace, where either the scale factor or a monotonic matter field functions to mark time. Lodging an objection in this context costs the past-eternalist the very models he needs. Second, even granting that the global context is timeless, internal relational time is still available by correlating a quantum degree of freedom that serves as a “clock” with the rest of the system. This construction, due to Don Page and William Wootters, conditions the static global state on sequential readings of such a “clock.” An ordinary Schrödinger evolution for the relative states can then be derived from the correlation. Karel Kuchař and others have raised two objections, however. One is the requirement in canonical general relativity that all observables commute with the Hamiltonian, which leads to static physical states that eliminate any time-dependence, reinstating quantum gravity’s problem of time. The other is that neither the correct propagators nor the correct two-time correlations are provided by the Page–Wootters mechanism. It gets “stuck” after the first time measurement. Solutions have, however, been found for these difficulties. Giovannetti, Lloyd, and Maccone have shown that these and additional objections can be resolved by constructing clock readings that are properly covariant and yield the right propagators and measurement statistics. In any model in which the rest of the cosmos keeps track of a suitable variable, the clock condition emerges as a structural feature rather than requiring independent postulation.20 The final reason is that the denial self-destructs. Without internal time, the cosmic seed has no succession of states at all, which means that no expansion can emerge from the seed, and cosmogenesis, past-eternal or otherwise, is missing from the model altogether. The irony is that the very idea of a past-eternal universe contains the notion of before-and-after that the internal-time assumption of the theorem makes explicit. To deny there is such a clock does not rescue an eternal past so much as prevent the question of whether there is one from even being asked.
The second assumption, as noted, derives from the very job that the past-eternalist is asking the seed to do. The essential condition is that the seed must be productive, which entails that some of its spectral weight be distributed over the continuum of expanding geometries into which it will evolve. The seed cannot be stationary under this condition. But it is also the case that, to have been waiting, its Hamiltonian must be bounded below. A seed that sleeps eternally before awakening must have a floor on which to rest. A past-eternal universe that later expands must have rested in its seed configuration over all early clock times. To reject this picture is to forsake the third road, Hav = 0, and its past-eternal seed altogether. The scope of the spectral theorem thus covers exactly those models in which there is a fact of the matter regarding what precedes the expansion of the universe. The past-eternalist cannot step outside these conditions into a universe in which there is no before-and-after, for without such a distinction there is no past at all, eternal or otherwise.
Residual improbabilities
The no-go theorem is the final wall in the prison we have been building for past-eternality, and any remaining escape routes must be catalogued, because the strength of the prison is judged by its weakest wall.
Only one of the static-seed models is precluded from being past-eternal as a corollary of the spectral no-go theorem, and then only on one explicitly stated premise. This is the simple harmonic universe, with its explicit decay channel, where it must be assumed, as resonance theory indicates, that the exhibited channel corresponds to a continuous spectral component of the seed. The strongest straightforward application of the spectral theorem, as distinct from a corollary to it, is to the emergent-universe model of George Ellis and Roy Maartens. The Ellis–Maartens model postulates a quasi-static seed of the kind covered by the theorem, with the instability of this seed independently confirmed by Aguirre and Kehayias’s wave-packet analysis.21 Other less obvious applications of the theorem, requiring judicious deployment because the cases are not as straightforward, involve models like those of Aguirre and Gratton in which universes emerge from a time-symmetric boundary,22 and Sean Carroll and Jennifer Chen’s minimal-entropy slice model.23 These models slip past the BGV theorem by a secret passageway. Aguirre–Gratton and Carroll–Chen both find a central surface at which the arrow of time reverses. Each model has a productive, nonstationary configuration of the kind the no-go theorem covers. Verifying the theorem’s applicability nonetheless requires checking that the relevant states couple to the continuum, have an absolutely continuous spectrum, and have a Hamiltonian that is bounded below. It is eminently plausible that these conditions are met and that the spectral theorem applies, but it has not yet been proved, so a definitive pronouncement would be premature.
In an earlier endnote, we mentioned the pure mathematical possibility of a “Cantor-dust intermediate” that remains to be considered here. In this case, rather than an absolutely continuous spectrum, we have the curious measure-theoretic case of a purely singular continuous seed. For this scenario, Wiener’s theorem on spectral averages guarantees only that time-averaged survival tends to zero.24 This still allows a flickering seed that returns near its original state with increasing rarity, but a nonvanishing presence in the asymptotic time average over an eternal past is denied. The result is weaker, but it still does the job because a presence with an average of zero over eternity is not the abiding one needed; it is an absence plagued by apparitions. Regardless, no actual cosmological model has a spectrum like this, so mentioning it is more a matter of mathematical than of physical completeness.
The ingenuity of a determined past-eternalist is considerable, so it is worth pricing the avoidance of a beginning. While exact stationary ground states are eternal, they are also impotent. More substantial proposals for cyclic recurrences lead to entropic costs that can only be paid down in scale-factor coin, placing them in debtor’s prison with BGV or Kinney–Stein holding the note. The last escape abandons quantum dynamics altogether and lies outside known physics. No theory describes it and no evidence supports it. None of these options have a past that remotely resembles the past of our universe. The prison we have built still has hidden passageways to a postern gate, but it opens on a cliff.
Coda
So it is that our universe finds itself in temporal bondage with eternity beyond its grasp. Given a positive average expansion rate, the BGV theorem shows that the past must be finite, thereby moving what was once a purely metaphysical question into an investigable physical realm. Our inflating world has a past boundary. The second road, a negative average expansion rate, which characterizes a contracting universe and leaves open the prospect that our world began in a bounce, is temporally bound by cords that range from the classical singularity theorems to the results of Tolman, Wall, and Kinney–Stein. The only remaining escape, the road along which our universe might have emerged from a past-eternal cosmic seed, is also bound by the mutual exclusivity of past-eternality and productivity demonstrated by the spectral no-go theorem. While this last result is predicated on internal time, the very posing of the question whether the past is eternal requires it as a premise. Thus, as far as present physics reaches, the trichotomy in Hav is closed along each of its roads. We are temporally bound on every side, yet here we are.
Lest we be tempted to credit this discussion with more than it has shown, we review what has not been delivered. BGV locates an edge beyond which lies mystery. Its sole condition is used to demonstrate that the inflationary account cannot be extended indefinitely into the past, but this does not, by itself, show that a past-eternal productive universe is impossible, nor does it, by itself, settle any theological question. Vilenkin has been quite adamant on this latter point,25 and the same caution applies to the former claim. Still, the BGV result is remarkable: with a simple inequality, almost no further assumptions, and an extraordinarily short argument, the prospect that the expanding past ends in a beginning becomes impossible to wish away. The general conclusion for the two other roads follows from the results assembled above. No result is sufficient to deliver this conclusion on its own, and the question of whether and how it all began is to be found at the end of past-directed worldlines along all three roads. With the principal known exits catalogued and every known possibility, however unlikely, taken into account, a simply stated conclusion emerges: any universe that was ever going to do something could not have been doing it forever already.
Appendix: The spectral no-go theorem
The full theorem is developed in my companion paper, “A Spectral No-Go Theorem for Past-Eternal Cosmological Seeds.”26 The architecture of the argument is as follows.
Consider a separable Hilbert space ℋ, containing a normalized vector, ψ. We specify a Hamiltonian operator, H, not to be confused with the Hubble parameter, as a self-adjoint operator that is bounded below so that the system’s ground sector is stable. Setting the ground energy to zero ensures the spectrum falls in the [0, ∞) range. The probability measure μψ(B) = ⟨ψ,EH(B)ψ⟩ on [0, ∞), where B is simply a range of energies, is then a spectral measure of ψ. What interests us is the survival amplitude of ψ, denoted by A(t), and given by the expression
A(t) = ⟨ψ,e−iHtψ⟩ = ∞∫0e−iEt dμψ(E).
Those familiar with Fourier analysis will recognize this as the Fourier transform of μψ from which the survival probability, S(t), is calculated as S(t) = A(t)2. This survival probability tells us how likely it is that a system prepared in state ψ will be found in ψ again after time t has passed.
Two results from quantum decay theory govern survival probabilities over long times. The first is a theorem proved in 1947 by Vladimir Fock and Nikolay Krylov showing that when μψ is absolutely continuous and dμψ(E) = ρ(E) dE, with the density function ρ Lebesgue integrable on [0, ∞), the Riemann–Lebesgue lemma, which states that the Fourier transform of an integrable function approaches zero as the frequency goes to infinity, ensures that A(t)→0, and thus S(t)→0, as t→∞. When their spectral weight lies at isolated energies, however, states never permanently lose their overlap with the state in which they started.
The Fock–Krylov result was refined by Leonid Khalfin in 1958, who proved that the decay of the survival amplitude was gentler than expected. Given that the Hamiltonian’s spectrum is bounded below, that amplitude, A(t), will extend on the lower half-plane to a bounded analytic function, yielding
∞∫−∞|lnA(t)|1 + t2 dt<∞
by the Paley–Wiener theorem.27 The decay thus declines sub-exponentially (since a decay even as fast as e−γt, for any positive constant γ, would violate the inequality) and has the generic form of a power law. This second theorem adds finesse to the first, but the result is garnish because the no-go theorem only requires that S(∞) = 0.
The Hermitian symmetry of A(t), that is, the fact that A(−t) = A(t)*, makes A(t)2 and hence S(t) even, namely, S(−t) = S(t). This means that regardless of whether we are moving into the future or into the past, the decay behaves exactly the same. The evenness of the survival probability is a fact about the intrinsic amplitude of the state, not about how the state was prepared, which is what also makes the no-go theorem relevant to evaluating the time-reversal models of Aguirre–Gratton and Carroll–Chen.
The no-go theorem holds whether we are inside or outside of homogeneous minisuperspace. All that differs between “inside” and “outside” is how a “clock” enters the picture. Either a monotonically changing scale factor or a monotonic matter field can serve as a clock internal to minisuperspace. Labeling the clock’s readings by τ gives order to the Wheeler–DeWitt dynamics by converting the frozen constraint into a unitary Schrödinger evolution, U(τ) = e−iHτ, with a self-adjoint reduced Hamiltonian, H, that is bounded below. Under these conditions, any nonstationary cosmic seed ψ with absolutely continuous spectral support will have a survival probability that goes to zero as τ→∞. This means that the probability the seed will be found in its initial quasi-static state at ever earlier clock times dwindles to nothing. In other words, the cosmic seed has no footprint in the remote past and therefore cannot serve as a past-eternal origin for the universe.
The fidelity of the cosmic seed picture is the casualty here, not the unitary history of the quantum system itself. Emergent-universe proposals assert the persistence of cosmic seeds through an infinite past, but this is exactly what cannot be true. In my technical paper developing this spectral theorem, I show why, even though a unitary history persists, a productive seed phase cannot be past-eternal. My demonstration invokes the RAGE (Ruelle–Amrein–Georgescu–Enss) theorem to prove that the productive component of the state cannot be confined to any bounded seed sector as clock times become arbitrarily remote. Any productive component escapes both past and future confinement. This escape is pointwise when the spectrum is absolutely continuous, whereas it holds in the time-average in the singular continuous case.
In contrast with this, outside minisuperspace, the existence of a clock has to be made an explicit premise. To satisfy this need, the model introduces a degree of freedom C registering the value τ, so that when the state ψ(τ) is conditioned on it, its evolution is unitary as governed by a self-adjoint relational Hamiltonian that is bounded below. This is Giovannetti, Lloyd, and Maccone’s covariant version of the original Page–Wootters construction in which time and dynamics emerge from the correlations between a clock subsystem and everything else in a globally stationary universe. Once these details are in place, we see that a nonstationary seed with absolutely continuous spectral support leads to the same conclusion in this context as well: a productive seed cannot constitute a past-eternal origin for the evolving universe.
Finally, these two cases, inside and outside, are bound together by the productivity dichotomy. Consider what happens when we decompose the quantum state of the cosmic seed ψ into its pure-point, absolutely continuous, and singular continuous parts, i.e., ψ = ψpp + ψac + ψsc. Taking these in order, what we find is that the pure-point part is governed by the quantum recurrence theorem, which makes its evolution almost periodic, returning arbitrarily close to ψpp infinitely often,28 meaning that it cannot irreversibly decay and its survival probability does not tend to zero. The absolutely continuous portion, ψac, whose survival probability is governed by the Fock–Krylov theorem, gives rise to irreversible transitions into the continuum of expanding geometries. According to Fock–Krylov, the survival probability of this productive component tends to zero as it moves to increasingly remote times. This does not prohibit a bound-state remnant of indefinite persistence, but any such remainder is inert. The state’s productive part has dissipated. Any eternal component of ψ must belong to this impotent remainder as an otiose leftover; it could never be an eternal origin. Lastly, a singular continuous part, ψsc, were it also present, while not necessarily vanishing pointwise, would by Wiener’s theorem have a contribution to survival that vanishes in the mean. The same conclusion as for ψac holds for ψsc in its time-averaged form. Stationary eigenstates, while eternal, are thus inert, and a superposition of bound states, while also eternal, has a sterile recurrence. The only cosmic seed states relevant to our past are productive states, every one of which is coupled to the continuum. They are thus the very ones that the spectral theorem prevents from being eternal. Instability and noneternality go hand in hand and are together the price of productivity.29
